In an independent line of work in mathematical physics, H. P. McKean in 1966 [1]} asked that in Boltzmann's \(H\) -Theorem, whether the derivatives of \(H(u)\)  alternate in signs. His question was finally disproved by E. Lieb [2]} in 1982. McKean also studied the higher order derivatives of \(h(Y_t)\)  with the motivation to generalize the maximal entropy principle to higher order derivatives of \(h(Y_t)\) . He showed the signs of the first two derivatives but failed to further show the signs of the third and fourth derivatives.
Surprisingly, the question on heat equation was never studied afterwards and almost lost in the literature [3]}.
